Note that the result of posterior convergence rate is necessary to establish \(||\Sigma _{\mathcal {Z}}^*-\Sigma _{\mathcal {Z}}||_{\operatorname{F}}=\operatorname{O}_p(\eta _n)\) , which is crucial to prove (22), where \(\eta _n=\sqrt{p}\epsilon _n\) . This implies that the posterior and true covariance matrix \(\Sigma _0\)  are concentrated around the projection of true covariance matrix \(\Sigma _0\)  onto the model \(\mathcal {Z}\)  at rate \(\eta _n\)  similar to remark of (3.15) in [1]}. A similar argument was also made in [1]}. Using auxiliary results in the supplementary material, the error due to Laplace approximation tends to zero with probability tending to one if \((p+\#\mathcal {Z})^2\eta _n=o(1)\)  under regular conditions. This gives Theorem REF .
